According to Maxwell's equations, accelerating charges emit electromagnetic radiation.
According to Quantum physics, heating causes electromagnetic radiation too.
Lasers emit bulk electromagnetic radiation which is described by quantum physics but have nothing to do with heat.
These 2 radiations, are they different kinds of radiation?
Your classification with respect to heat and acceleration is irrelevant .
All are the same radiation which after emission follows classical electrodynamics when in bulk.
As the other answers state heat is vibrating molecules but heat is in the median between classical and quantum formulations. It was actually the black body radiation , a classical thermodynamic problem which necessitated the existence of quanta of light, i.e. photons, from the E=h*nu identity of the photons where E is the energy and nu the frequency and h Planck's constant.
The vibrating molecules are in quantized states, except the potentials are so shallow that they are practically a continuum. It is necessary though to assume the quantization in order to avoid the ultraviolet catastrophe .
As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths. The black-body radiation graph is also compared with the classical model of Rayleigh and Jeans.
The classical goes off to infinity as the wavelength decreases .
One has to understand that the underlying framework of classical physics, the classical theories that fit macroscopic data well, (except in special calculations as in black body radiation or very low intensities or lasers or transistors etc), is the quantum state of matter . At microscopic distances everything is quantized, even the photons from accelerating charges. What happens is that for macroscopic values and large ensembles the classical behavior emerges : thermodynamics from quantum statistical mechanics, for example. In a similar way large ensembles of photons are completely consistent in mathematical behavior with the solutions of Maxwell's equations. If you are interested you could check this blog essay on how the classical emerges from the individual photons.